Lifting Functors from Pos to Pries

Jim de Groot

The Australian National University College of Engeneering & Computer Science

TACL 19 June 2019

1 / 14 Contents

Coalgebraic positive logic Predicate liftings ▸ Connection▸ between Pos- and Pries-functors Lifting via semantics (predicate liftings) ▸ Lifting via (coﬁltered) limits ▸ Comparison ▸ ▸

2 / 14 A coalgebra morphism f X , γ X ′, γ′ is a morphism

▸ X∶ ( ) → X( ′ ) γ γ′ TX TX ′

Coalgebras

Deﬁnition A coalgebra for a functor T C C is a pair X , γ where X C and γ X TX . ▸ ∶ → ( ) ∈ ∶ →

3 / 14 Coalgebras

Deﬁnition A coalgebra for a functor T C C is a pair X , γ where X C and γ X TX . ▸ ∶ → ( ) A coalgebra morphism f X , γ X ′, γ′ is a morphism ∈ ∶ → ▸ X∶ ( ) → X( ′ ) γ γ′ TX TX ′

3 / 14 Coalgebras

Deﬁnition A coalgebra for a functor T C C is a pair X , γ where X C and γ X TX . ▸ ∶ → ( ) A coalgebra morphism f X , γ X ′, γ′ is a morphism ∈ ∶ → ▸ X∶ ( f ) → X( ′ ) γ γ′ TX Tf TX ′

3 / 14 An n-ary predicate lifting is a natural transformation

λ Upn Up T.

For a set Λ of predicate liftings,∶ → deﬁne○ L T, Λ by

λ ϕ p ϕ ϕ ϕ ϕ ( ϕ1,) . . . , ϕn .

Interpretation∶∶= in S with⊺ S valuationS ∨ S V∧ PropS ♡ ( : )

λ x ϕ1, . . . , ϕn γ ∶x λ →ϕ1 ,..., ϕn . J K J K ⊩ ♡ ( ) ⇐⇒ ( ) ∈ ( )

Deﬁnition

Coalgebraic positive logic

Setting Up T Pos DL

4 / 14 For a set Λ of predicate liftings, deﬁne L T, Λ by

λ ϕ p ϕ ϕ ϕ ϕ ( ϕ1,) . . . , ϕn .

Interpretation∶∶= in S with⊺ S valuationS ∨ S V∧ PropS ♡ ( : )

λ x ϕ1, . . . , ϕn γ ∶x λ →ϕ1 ,..., ϕn . J K J K ⊩ ♡ ( ) ⇐⇒ ( ) ∈ ( )

Coalgebraic positive logic

Setting Up T Pos DL

Deﬁnition An n-ary predicate lifting is a natural transformation

λ Upn Up T.

∶ → ○

4 / 14 Interpretation in with valuation V Prop :

λ x ϕ1, . . . , ϕn γ ∶x λ →ϕ1 ,..., ϕn . J K J K ⊩ ♡ ( ) ⇐⇒ ( ) ∈ ( )

Coalgebraic positive logic

Setting Up T Pos DL

Deﬁnition An n-ary predicate lifting is a natural transformation

λ Upn Up T.

For a set Λ of predicate liftings,∶ → deﬁne○ L T, Λ by

λ ϕ p ϕ ϕ ϕ ϕ ( ϕ1,) . . . , ϕn .

∶∶= S ⊺ S S ∨ S ∧ S ♡ ( )

4 / 14 Coalgebraic positive logic

Setting Up T Pos DL

Deﬁnition An n-ary predicate lifting is a natural transformation

λ Upn Up T.

For a set Λ of predicate liftings,∶ → deﬁne○ L T, Λ by

λ ϕ p ϕ ϕ ϕ ϕ ( ϕ1,) . . . , ϕn .

Interpretation∶∶= in SX⊺,S γ Swith∨ valuationS ∧ S V♡ (Prop Up)X :

λ x ϕ1(, . . . ,) ϕn γ x λ∶ ϕ1 ,...,→ ϕn . J K J K ⊩ ♡ ( ) ⇐⇒ ( ) ∈ ( ) 4 / 14 Coalgebraic positive logic

Setting ClpUp T Pries DL

Deﬁnition An n-ary predicate lifting is a natural transformation

λ ClpUpn ClpUp T.

For a set Λ of predicate∶ liftings,→ deﬁne L○T, Λ by

λ ϕ p ϕ ϕ ϕ ϕ ( ϕ1,) . . . , ϕn .

Interpretation∶∶= in S ⊺,S γ Swith∨ valuationS ∧ S ♡V (Prop ClpUp) :

λ x ϕ1(,X . . . ,) ϕn γ x λ ∶ ϕ1 ,...,→ ϕn X. J K J K ⊩ ♡ ( ) ⇐⇒ ( ) ∈ ( ) 4 / 14 Intuition In a c -coalgebras X , γ , γ maps x to its set of successors.

PredicateP liftings ( ) ◻ n Deﬁne λ , λ Up Up c by

◻ ∶ → ○ P n λX a b c X b a , λX a b c X b a .

λ◻ ◻ Then:( x) = { ∈ϕP iﬀ S γ⊆x } λX ϕ ( ) iﬀ= { γ∈ xP Sϕ ∩. ≠ ∅} J K J K ⊩ ♡ ( ) ∈ ( ) ( ) ⊆

Example: Convex powerset functor on Pos

Pc ∶ Pos → Pos X convex subsets of X ordered by :

▸ ↦ a b iﬀ a b and⊑ b a

′ For f X X deﬁne⊑ c f by ⊆ ↓c f a ⊆f↑ a f a

▸ ∶ → P (P )( ) = ↓ [ ] ∩ ↑ [ ]

5 / 14 Predicate liftings ◻ n Deﬁne λ , λ Up Up c by

◻ ∶ → ○ P n λX a b c X b a , λX a b c X b a .

λ◻ ◻ Then:( x) = { ∈ϕP iﬀ S γ⊆x } λX ϕ ( ) iﬀ= { γ∈ xP Sϕ ∩. ≠ ∅} J K J K ⊩ ♡ ( ) ∈ ( ) ( ) ⊆

Example: Convex powerset functor on Pos

Pc ∶ Pos → Pos X convex subsets of X ordered by :

▸ ↦ a b iﬀ a b and⊑ b a

′ For f X X deﬁne⊑ c f by ⊆ ↓c f a ⊆f↑ a f a

Intuition▸ ∶ → P (P )( ) = ↓ [ ] ∩ ↑ [ ] In a c -coalgebras X , γ , γ maps x to its set of successors.

P ( )

5 / 14 λ◻ ◻ Then: x ϕ iﬀ γ x λX ϕ iﬀ γ x ϕ . J K J K ⊩ ♡ ( ) ∈ ( ) ( ) ⊆

Example: Convex powerset functor on Pos

Pc ∶ Pos → Pos X convex subsets of X ordered by :

▸ ↦ a b iﬀ a b and⊑ b a

′ For f X X deﬁne⊑ c f by ⊆ ↓c f a ⊆f↑ a f a

Intuition▸ ∶ → P (P )( ) = ↓ [ ] ∩ ↑ [ ] In a c -coalgebras X , γ , γ maps x to its set of successors.

PredicateP liftings ( ) ◻ n Deﬁne λ , λ Up Up c by

◻ ∶ → ○ P n λX a b c X b a , λX a b c X b a . ( ) = { ∈ P S ⊆ } ( ) = { ∈ P S ∩ ≠ ∅} 5 / 14 Example: Convex powerset functor on Pos

Pc ∶ Pos → Pos X convex subsets of X ordered by :

▸ ↦ a b iﬀ a b and⊑ b a

′ For f X X deﬁne⊑ c f by ⊆ ↓c f a ⊆f↑ a f a

Intuition▸ ∶ → P (P )( ) = ↓ [ ] ∩ ↑ [ ] In a c -coalgebras X , γ , γ maps x to its set of successors.

PredicateP liftings ( ) ◻ n Deﬁne λ , λ Up Up c by

◻ ∶ → ○ P n λX a b c X b a , λX a b c X b a .

λ◻ ◻ Then:( x) = { ∈ϕP iff S γ⊆x } λX ϕ ( ) iff= { γ∈ xP Sϕ ∩. ≠ ∅} J K J K 5 / 14 ⊩ ♡ ( ) ∈ ( ) ( ) ⊆ Predicate liftings ◻ n Define λ , λ ClpUp ClpUp c by

◻ n λX a b ∶ c b→ a ,○ V λX a b c b a .

λ◻ ◻ Then:( x) = { ∈ϕV Xiﬀ S γ⊆x } λX ϕ ( )iﬀ= { γ∈ xV X Sϕ ∩. ≠ ∅} J K J K ⊩ ♡ ( ) ∈ ( ) ( ) ⊆

Example: Convex Vietoris functor on Pries

Vc ∶ Pries → Pries closed convex subsets of ordered by , topologised by

▸ X ↦ a b c b a , X a b c ⊑ b a ,

where} a= {ClpUp∈ V X .S ⊆ } } = { ∈ V X S ∩ ≠ ∅} For a morphism f ′ let f b f b f b . ∈ X c ▸ ∶ X → X (V )( ) = ↓ [ ] ∩ ↑ [ ]

6 / 14 Example: Convex Vietoris functor on Pries

Vc ∶ Pries → Pries closed convex subsets of ordered by , topologised by

▸ X ↦ a b c b a , X a b c ⊑ b a ,

where} a= {ClpUp∈ V X .S ⊆ } } = { ∈ V X S ∩ ≠ ∅} For a morphism f ′ let f b f b f b . ∈ X c ▸ ∶ X → X (V )( ) = ↓ [ ] ∩ ↑ [ ] Predicate liftings ◻ n Deﬁne λ , λ ClpUp ClpUp c by

◻ n λX a b ∶ c b→ a ,○ V λX a b c b a .

λ◻ ◻ Then:( x) = { ∈ϕV Xiﬀ S γ⊆x } λX ϕ ( )iﬀ= { γ∈ xV X Sϕ ∩. ≠ ∅} J K J K ⊩ ♡ ( ) ∈ ( ) ( ) ⊆ 6 / 14 Let DT,Λ be the sub-DL of Up T W generated by

▸ X λWX a1,..., an λ( Λ(, aiX ))ClpUp .

For f { let( DT,Λf DT)S,Λ ∈ DT,Λ∈ be theX } restriction of Up T Wf Tf −1. ▸ ∶ X → Y ∶ Y → X ( ( )) = ( )

Deﬁnition For a set Λ of predicate liftings, deﬁne DT,Λ Pries DL by:

∶ →

Semantic functor lift

Setting W

Up pf T Pos DL Pries Tˆ ClpUp

7 / 14 For f let DT,Λf DT,Λ DT,Λ be the restriction of Up T Wf Tf −1. ▸ ∶ X → Y ∶ Y → X ( ( )) = ( )

Semantic functor lift

Setting W

Up pf T Pos DL Pries Tˆ ClpUp

Deﬁnition For a set Λ of predicate liftings, deﬁne DT,Λ Pries DL by: Let D be the sub-DL of Up T W generated by T,Λ ∶ →

▸ X λWX a1,..., an λ( Λ(, aiX ))ClpUp .

{ ( )S ∈ ∈ X }

7 / 14 Semantic functor lift

Setting W

Up pf T Pos DL Pries Tˆ ClpUp

Deﬁnition For a set Λ of predicate liftings, deﬁne DT,Λ Pries DL by: Let D be the sub-DL of Up T W generated by T,Λ ∶ →

▸ X λWX a1,..., an λ( Λ(, aiX ))ClpUp .

For f { let( DT,Λf DT)S,Λ ∈ DT,Λ∈ be theX } restriction of Up T Wf Tf −1. ▸ ∶ X → Y ∶ Y → X ( ( )) = ( )

7 / 14 Semantic functor lift

Setting W

Up pf T Pos DL Pries Tˆ ClpUp Deﬁnition For a set Λ of predicate liftings, deﬁne DT,Λ Pries DL by: Let D be the sub-DL of Up T W generated by T,Λ ∶ →

▸ X λWX a1,..., an λ( Λ(, aiX ))ClpUp .

For f { let( DT,Λf DT)S,Λ ∈ DT,Λ∈ be theX } restriction of Up T Wf Tf −1. ▸ ∶ X → Y ∶ Y → X Deﬁne the( semantic( )) = lift( of) T w.r.t. Λ by

Tˆ pf DT,Λ Pries Pries. 7 / 14 = ○ ∶ → Proof idea

For a Priestley space , we have DPc ,Λ ClpUp c via

ϕ ClpUp c DPc ,Λ generated by X X = (V X ) ∶ (V Xϕ) → a λX◻ a , ϕ a λn a .

( } ) = ( ) (} ) = ( )

Example

◻ n Consider c and Λ λ , λ . Then

P = { }c Λ c .

(ÅP ) = V

8 / 14 Example

◻ n Consider c and Λ λ , λ . Then

P = { }c Λ c . Å Proof idea (P ) = V

For a Priestley space , we have DPc ,Λ ClpUp c via

ϕ ClpUp c DPc ,Λ generated by X X = (V X ) ∶ (V Xϕ) → a λX◻ a , ϕ a λn a .

( } ) = ( ) (} ) = ( )

8 / 14 Remarks

1. Posf Priesf 2. For Pries we have ClpUp Up W . ≅ f 3. For Pries, let Pries be the coslice category and X ∈ f X = ( X ) UX Priesf Pries the obvious forgetful functor. X ∈ X ↓ Then lim UX . ∶ (X ↓ ) → X =

Functor lift using coﬁltered limits

Setting W

Up pf T Pos DL Pries T ClpUp

9 / 14 Functor lift using coﬁltered limits

Setting W

Up pf T Pos DL Pries T ClpUp Remarks

9 / 14 Functor lift using coﬁltered limits

Objects

Suppose T Pos Pos restricts to an endofunctor on Posf . For Pries deﬁne ∶ → T lim TUX . X ∈ X = ( )

10 / 14 Functor lift using coﬁltered limits

Objects

Suppose T Pos Pos restricts to an endofunctor on Posf . For Pries deﬁne ∶ → T lim TUX . X ∈ X = ( )

X h i j

X X

10 / 14 Functor lift using coﬁltered limits

Objects

Suppose T Pos Pos restricts to an endofunctor on Posf . For Pries deﬁne ∶ → T lim TUX . X ∈ X = ( ) T

X Th T i T j

X X

10 / 14 Functor lift using coﬁltered limits

Objects

Suppose T Pos Pos restricts to an endofunctor on Posf . For Pries deﬁne ∶ → T lim TUX . X ∈ X = ( ) T

X Th T i T j

X X

10 / 14 Therefore T is a cone for TUY . By the limit property we get Tf T lim TUY T . X ∶ X → ( ) = Y

X Y h i j

Y Y

Functor lift using coﬁltered limits

Objects

Suppose T Pos Pos restricts to an endofunctor on Posf . For Pries deﬁne ∶ → T lim TUX . X ∈ Morphisms X = ( )

If f , every object in the diagram UY is also inU X .

∶ X → Y

10 / 14 Therefore T is a cone for TUY . By the limit property we get Tf T lim TUY T . X ∶ X → ( ) = Y

Functor lift using coﬁltered limits

Objects

Suppose T Pos Pos restricts to an endofunctor on Posf . For Pries deﬁne ∶ → T lim TUX . X ∈ Morphisms X = ( )

If f , every object in the diagram UY is also inU X .

∶ X → Y f

X Y h i j

Y Y

10 / 14 Functor lift using coﬁltered limits

Objects

Suppose T Pos Pos restricts to an endofunctor on Posf . For Pries deﬁne ∶ → T lim TUX . X ∈ Morphisms X = ( )

If f , every object in the diagram UY is also inU X .

∶ X → Y f

X Y h i j

Y Y Therefore T is a cone for TUY . By the limit property we get Tf T lim TUY T . X 10 / 14 ∶ X → ( ) = Y 1. The collection s sX X ∈Pries ClpUp T Up T W is a natural transformation. = ( ) ∶ ○ → ○ ○ 2. If T preserves epis and coﬁltered limits then sX is injective.

3.D T,Λ is a subframe of im sX . 4. If T is embedding-preserving then D im s . X T,Λ X X ≅

We know that is a cone for UX .

Therefore Up TW is a cocone of Up TWUX . ▸ X By deﬁnition ClpUp T colim Up TWUX , so there is a ▸ ( X ) ( ) homomorphism ▸ ( X ) = ( ( )) sX ClpUp T Up TW

Lemma ∶ ( X ) → ( X )

Comparison of functor lifts

View ClpUp T as sub-DL of Up TW .

( X ) ( X )

11 / 14 1. The collection s sX X ∈Pries ClpUp T Up T W is a natural transformation. = ( ) ∶ ○ → ○ ○ 2. If T preserves epis and coﬁltered limits then sX is injective.

3.D T,Λ is a subframe of im sX . 4. If T is embedding-preserving then D im s . X T,Λ X X ≅

Therefore Up TW is a cocone of Up TWUX .

By deﬁnition ClpUp T colim Up TWUX , so there is a ▸ ( X ) ( ) homomorphism ▸ ( X ) = ( ( )) sX ClpUp T Up TW

Lemma ∶ ( X ) → ( X )

Comparison of functor lifts

View ClpUp T as sub-DL of Up TW . We know that is a cone for U . ( X ) ( X X ) ▸ X

11 / 14 1. The collection s sX X ∈Pries ClpUp T Up T W is a natural transformation. = ( ) ∶ ○ → ○ ○ 2. If T preserves epis and coﬁltered limits then sX is injective.

3.D T,Λ is a subframe of im sX . 4. If T is embedding-preserving then D im s . X T,Λ X X ≅

By deﬁnition ClpUp T colim Up TWUX , so there is a homomorphism ▸ ( X ) = ( ( )) sX ClpUp T Up TW

Lemma ∶ ( X ) → ( X )

Comparison of functor lifts

View ClpUp T as sub-DL of Up TW . We know that is a cone for U . ( X ) ( X X ) Therefore Up TW is a cocone of Up TWUX . ▸ X ▸ ( X ) ( )

11 / 14 1. The collection s sX X ∈Pries ClpUp T Up T W is a natural transformation. = ( ) ∶ ○ → ○ ○ 2. If T preserves epis and coﬁltered limits then sX is injective.

3.D T,Λ is a subframe of im sX . 4. If T is embedding-preserving then D im s . X T,Λ X X ≅

Lemma

Comparison of functor lifts

View ClpUp T as sub-DL of Up TW . We know that is a cone for U . ( X ) ( X X ) Therefore Up TW is a cocone of Up TWUX . ▸ X By deﬁnition ClpUp T colim Up TWUX , so there is a ▸ ( X ) ( ) homomorphism ▸ ( X ) = ( ( )) sX ClpUp T Up TW

∶ ( X ) → ( X )

11 / 14 2. If T preserves epis and coﬁltered limits then sX is injective.

3.D T,Λ is a subframe of im sX . 4. If T is embedding-preserving then D im s . X T,Λ X X ≅

Comparison of functor lifts

Lemma ∶ ( X ) → ( X )

1. The collection s sX X ∈Pries ClpUp T Up T W is a natural transformation. = ( ) ∶ ○ → ○ ○

11 / 14 3.D T,Λ is a subframe of im sX . 4. If T is embedding-preserving then D im s . X T,Λ X X ≅

Comparison of functor lifts

Lemma ∶ ( X ) → ( X )

1. The collection s sX X ∈Pries ClpUp T Up T W is a natural transformation. = ( ) ∶ ○ → ○ ○ 2. If T preserves epis and coﬁltered limits then sX is injective.

11 / 14 4. If T is embedding-preserving then DT,Λ im sX .

X ≅

Comparison of functor lifts

Lemma ∶ ( X ) → ( X )

1. The collection s sX X ∈Pries ClpUp T Up T W is a natural transformation. = ( ) ∶ ○ → ○ ○ 2. If T preserves epis and coﬁltered limits then sX is injective.

3.D T,Λ is a subframe of im sX .

11 / 14 Comparison of functor lifts

Lemma ∶ ( X ) → ( X )

1. The collection s sX X ∈Pries ClpUp T Up T W is a natural transformation. = ( ) ∶ ○ → ○ ○ 2. If T preserves epis and coﬁltered limits then sX is injective.

3.D T,Λ is a subframe of im sX . 4. If T is embedding-preserving then D im s . X T,Λ X X ≅ 11 / 14 Example Finitely generated convex powerset functor on Pos.

Comparison of functor lifts

Theorem Let T be an endofunctor on Pos which

restricts to Posf ; and preserves epis, embeddings and coﬁltered limits. ▸ Let Λ be the set of all positive predicate liftings for T. Then there ▸ is a natural isomorphism T T.ˆ

→

12 / 14 Comparison of functor lifts

Theorem Let T be an endofunctor on Pos which

restricts to Posf ; and preserves epis, embeddings and coﬁltered limits. ▸ Let Λ be the set of all positive predicate liftings for T. Then there ▸ is a natural isomorphism T T.ˆ

Example → Finitely generated convex powerset functor on Pos.

12 / 14 Generalization?

Similar methods and result for lifting Set-functors to Stone-functors. ▸ More general approach?

▸

13 / 14 Thank you

14 / 14